Hilbert series for free lie superalgebras and related topics

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Abstract

We consider Hilbert series of ordinary Lie algebras, restricted (or p-) Lie algebras,
and color Lie (p-)superalgebras. We derive a dimension formula similar to a wellknown
Witt’s formula for free color Lie superalgebras and a certain class of color Lie
p-superalgebras. A Lie (super)algebra analogue of a well-known Schreier’s formula
for the rank of a subgroup of finite index in a free group was found by V. M. Petrogradsky.
In this dissertation, Petrogradsky’s formulas are extended to the case of
color Lie (p-)superalgebras. We establish more Schreier-type formulas for the ranks
of submodules of free modules over free associative algebras and free group algebras.
As an application, we consider Hopf subalgebras of some cocommutative Hopf algebras.
Also, we apply our version of Witt and Schreier formulas to study relatively
free color Lie (p-)superalgebras and to prove that the free color Lie superalgebra and
its enveloping algebra have the same entropy. Y. A. Bahturin and A. Y. Olshanskii
proved that the relative growth rate of a finitely generated subalgebra K of a free Lie
algebra L of finite rank is strictly less than the growth rate of the free Lie algebra
itself. We show that this theorem cannot be extended to free color Lie superalgebras
in general. However, we establish it in a special case.